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The Wiener Criterion for Nonlocal Dirichlet Problems

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Abstract

We study the boundary behavior of solutions to the Dirichlet problems for integro-differential operators with order of differentiability \(s \in (0, 1)\) and summability \(p>1\). We establish a nonlocal counterpart of the Wiener criterion, which characterizes a regular boundary point in terms of the nonlocal nonlinear potential theory.

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Correspondence to Se-Chan Lee.

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Minhyun Kim gratefully acknowledges financial support by the German Research Foundation (GRK 2235 - 282638148). The research of Ki-Ahm Lee is supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIP): NRF-2021R1A4A1027378. Se-Chan Lee is supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2022R1A6A3A01086546).

Appendix A. Algebraic Inequalities

Appendix A. Algebraic Inequalities

In this section, we provide some algebraic inequalities that are used in the Caccioppoli-type estimates. Let \(p \in (1, \infty )\) and \(\beta , \gamma \in \mathbb {R}\) be such that \(\beta = \gamma -(p-1)\). Recall that, in the local case, we use the following inequalities for the proof of Caccioppoli-type estimates [31, Theorem 8.25 and 8.26]: assume \(\beta \ne 0\), \(\gamma \ne 0\), and let \(l \ge 0\). Then, there exists constants \(c, C > 0\), depending only on p, such that

$$\begin{aligned} \begin{aligned} |\nabla u|^{p-2} \nabla u \cdot \nabla (f(u) \eta ^p)&\ge |\nabla F(u)|^p \eta ^p - \frac{|\gamma |}{|\beta |} |\nabla F(u)|^{p-1} \eta ^{p-1} |F(u)| |\nabla \eta | \\&\ge \frac{1}{p} \left( |\nabla F(u)|^p \eta ^p - \left( \frac{|\gamma |}{|\beta |} \right) ^p |F(u)|^{p} |\nabla \eta |^p \right) \\&\ge c \, |\nabla (F(u) \eta )|^p - C \left( 1+\left( \frac{|\gamma |}{|\beta |} \right) ^p \right) |F(u)|^p |\nabla \eta |^p, \end{aligned} \end{aligned}$$

where \(f, F: I \rightarrow [0, \infty )\) are defined by

$$\begin{aligned} f(t) = \frac{1}{\beta } \left( (t+d)^{\beta } - (l+d)^{\beta } \right) \quad \text {and}\quad F(t) = \frac{p}{\gamma } (t+d)^{\gamma /p}. \end{aligned}$$
(A.1)

Here, \(I = [l, \infty )\) when \(\beta > 0\) and \(I = [0, l]\) when \(\beta < 0\). As a discrete version, we prove the following algebraic inequality.

Lemma A.1

Assume that \(\beta \ne 0\) and \(\gamma \ne 0\). There exist \(c, C > 0\), depending only on p, such that

$$\begin{aligned} \begin{aligned}&|a-b|^{p-2}(a-b) (f(a) \eta _1^p - f(b) \eta _2^p) \\&\quad \ge c \left| F(a) \eta _1 - F(b) \eta _2 \right| ^p - C \left( 1+ \left( \frac{|\gamma |}{|\beta |} \right) ^p \right) \max \lbrace |F(a)|, |F(b)| \rbrace ^p |\eta _1-\eta _2|^p \end{aligned} \end{aligned}$$

for any \(a, b \in I\) and \(\eta _1, \eta _2 \ge 0\).

Some inequalities similar to Lemma A.1 are known for almost all values of \(\beta \). See, for instance, [5] for \(\beta \ge 1\) and [10] for \(\beta < -(p-1)\). Lemma A.1 can be proved in a similar way, but let us provide a proof to make the paper self-contained. To this end, we first prove the following two lemmas as intermediate steps.

Lemma A.2

If \(\beta \ne 0\) and \(\gamma \ne 0\), then

$$\begin{aligned} |a-b|^{p-2}(a-b)(f(a)-f(b)) \ge |F(a)-F(b)|^p \end{aligned}$$

and

$$\begin{aligned} |a-b|^{p-1} \min \left\{ F'(a), F'(b) \right\} ^{p-1} \le |F(a)-F(b)|^{p-1} \end{aligned}$$

for any \(a, b \in I\).

Proof

We may assume that \(a>b\). Then, we have

by Jensen’s inequality, and

since \(F'\) is positive on \([0, \infty )\). \(\square \)

Lemma A.3

Let \(A, B \in \mathbb {R}\) and \(\eta _1, \eta _2 \ge 0\), then

$$\begin{aligned} \begin{aligned} |A-B|^p \min \lbrace \eta _1, \eta _2 \rbrace ^p&\ge 2^{1-p} |A \eta _1 - B \eta _2|^p - \max \lbrace |A|, |B| \rbrace ^p |\eta _1-\eta _2|^p \quad \text {and} \\ |A-B|^p \max \lbrace \eta _1, \eta _2 \rbrace ^p&\le 2^{p-1} |A \eta _1 - B \eta _2|^p + 2^{p-1} \max \lbrace |A|, |B| \rbrace ^p |\eta _1-\eta _2|^p. \end{aligned} \end{aligned}$$

Proof

We may assume that \(\eta _1 \ge \eta _2\). Then, the desired inequalities follow from the equalities

$$\begin{aligned} A\eta _1 - B\eta _2 = (A-B) \eta _2 + A (\eta _1-\eta _2) = (A-B) \eta _1 + B (\eta _1-\eta _2) \end{aligned}$$

and the triangle inequality. \(\square \)

We prove Lemma A.1 by using Lemmas A.2 and A.3.

Proof of Lemma A.1

We may assume that \(a > b\). Then, we have

$$\begin{aligned}&(a-b)^{p-1} (f(a) \eta _1^p - f(b) \eta _2^p) \nonumber \\ {}&= (a-b)^{p-1}(f(a) - f(b))\eta _1^p + (a-b)^{p-1} f(b)(\eta _1^p-\eta _2^p) \end{aligned}$$
(A.2)
$$\begin{aligned} \nonumber \\ {}&= (a-b)^{p-1}(f(a) - f(b))\eta _2^p + (a-b)^{p-1} f(a)(\eta _1^p-\eta _2^p). \end{aligned}$$
(A.3)

We apply Lemma A.2 to (A.2) if \(\beta \ge 1\) or to (A.3) if \(\beta \le 1\). Then, we obtain

$$\begin{aligned} \begin{aligned} J :=&~ (a-b)^{p-1} (f(a) \eta _1^p - f(b) \eta _2^p) \\ \ge&~ |F(a)-F(b)|^p \min \lbrace \eta _1, \eta _2 \rbrace ^p \\ {}&-|F(a)-F(b)|^{p-1} \max \left\{ \frac{|f(a)|}{F'(a)^{p-1}}, \frac{|f(b)|}{F'(b)^{p-1}} \right\} |\eta _1^p-\eta _2^p|. \end{aligned} \end{aligned}$$

By using

$$\begin{aligned} \frac{|f(t)|}{F'(t)^{p-1}} \le \frac{|\gamma |}{p|\beta |} |F(t)|, \quad |\eta _1^p-\eta _2^p| \le p |\eta _1-\eta _2|\max \lbrace \eta _1, \eta _2 \rbrace ^{p-1}, \end{aligned}$$

and Young’s inequality, we deduce

$$\begin{aligned} \begin{aligned} J&\ge |F(a)-F(b)|^p \min \lbrace \eta _1, \eta _2 \rbrace ^p \\&\quad - \frac{1}{2^{2p-1}} |F(a)\!-\!F(b)|^p \max \lbrace \eta _1, \eta _2 \rbrace ^p \!-\! C \left( \frac{|\gamma |}{|\beta |} \right) ^p \max \lbrace |F(a)|, |F(b)| \rbrace ^p |\eta _1 \!-\! \eta _2|^p, \end{aligned} \end{aligned}$$

where \(C = C(p) > 0\). Applying Lemma A.3 with \(A = F(a)\) and \(B = F(b)\) finishes the proof. \(\square \)

The case \(\gamma = 0\), or equivalently \(\beta = -(p-1)\), is treated in the following lemma, which is a discrete version of

$$\begin{aligned} |\nabla u|^{p-2} \nabla u \cdot \nabla (f(u) \eta ^p) \ge c |\nabla \log (u+d)|^p \eta ^p - C |\nabla \eta |^p, \end{aligned}$$

where f is given by (A.1) with \(\beta = -(p-1)\) and \(c, C > 0\) are constants depending only on p.

Lemma A.4

Assume that \(\gamma = \beta + p-1 = 0\). There exist \(c, C > 0\), depending only on p, such that

$$\begin{aligned} \begin{aligned}{}&{} |a-b|^{p-2}(a-b) (f(a)\eta _1^p - f(b)\eta _2^p)\\{}&{} \quad \ge c\, |\log (a+d)- \log (b+d)|^p \min \lbrace \eta _1, \eta _2 \rbrace ^p - C |\eta _1-\eta _2|^p \end{aligned} \end{aligned}$$

for any \(a, b \in I\) and \(\eta _1, \eta _2 \ge 0\).

Proof

We may assume that \(a > b\). By applying [14, Lemma 3.1] to \(\eta _1\) and \(\eta _2\) with

$$\begin{aligned} \varepsilon = \delta \frac{a-b}{a+d} \in (0,1), \quad \delta \in (0, 1), \end{aligned}$$

we have

$$\begin{aligned} \eta _1^p - \eta _2^p \le C \varepsilon \eta _2^p + C \varepsilon ^{1-p} |\eta _1-\eta _2|^p, \end{aligned}$$

where \(C = C(p) > 0\). Thus, it follows from (A.3) and \(f(a) \ge \frac{1}{1-p}(a+d)^{1-p}\) that

$$\begin{aligned} \begin{aligned} K :=&~ (a-b)^{p-1}(f(a)\eta _1^p - f(b)\eta _2^p) \\ \ge&~ (a-b)^{p-1} (f(a)-f(b)) \eta _2^p - C\delta \left( \frac{a-b}{a+d} \right) ^p \eta _2^p - C \delta ^{1-p} |\eta _1-\eta _2|^p. \end{aligned} \end{aligned}$$

By using the same arguments as in the proof of Lemma A.2, we obtain

$$\begin{aligned} \left( \frac{a-b}{a+d} \right) ^{p} \le |\log (a+d) - \log (b+d)|^p \le (a-b)^{p-1}(f(a)-f(b)). \end{aligned}$$

Therefore, we deduce that

$$\begin{aligned} K \ge (1-C\delta ) |\log (a+d) - \log (b+d)|^p \eta _2^p - C \delta ^{1-p} |\eta _1-\eta _2|^p, \end{aligned}$$

from which we conclude the lemma by taking \(\delta \) sufficiently small. \(\square \)

We also provide the following algebraic inequality, which is similar to Lemma A.1.

Lemma A.5

Let \(a, b \in \mathbb {R}\), \(\eta _1, \eta _2 \ge 0\), and assume that \(\gamma \) satisfies (5.3). Let \(\tau = \gamma /(p-1)\) and \(q = p\gamma /(p-\gamma /(p-1))\), then

$$\begin{aligned} \begin{aligned}&|a-b|^{p-2}(a-b)(g(a_+) \eta _1^p - g(b_+) \eta _2^p) \\&\quad \ge c\, |G(a_+) \eta _1 - G(b_+)\eta _2|^p - C \max \lbrace (1+a_+)^{\gamma }, (1+b_+)^{\gamma } \rbrace \textbf{1}_{\lbrace a>0 \rbrace } \textbf{1}_{\lbrace b>0 \rbrace } |\eta _1 - \eta _2|^p, \end{aligned} \end{aligned}$$

where

$$\begin{aligned} g(t) = \frac{1}{\tau -1}(1-(1+t)^{1-\tau }), \quad G(t) = \frac{q}{\gamma }((1+t)^{\gamma /q}-1), \end{aligned}$$

and \(c, C > 0\) are constants depending only on p.

Proof

We may assume that \(a>b\). The same arguments as in the proof of Lemma A.2 show that

$$\begin{aligned} (a_+-b_+)^{p-1} (g(a_+) - g(b_+)) \ge |G(a_+) - G(b_+)|^p \end{aligned}$$
(A.4)

and

$$\begin{aligned} (a_+-b_+)^{p-1} g'(a_+)^{\frac{p-1}{p}} \le |G(a_+)-G(b_+)|^{p-1}. \end{aligned}$$

The case \(b \le 0\) follows from (A.4). Thus, we assume \(b > 0\). Since \(a-b \ge a_+ - b_+\), \((a-b)^{p-1} g(b_+) = (a_+ - b_+)^{p-1} g(b_+)\), and \(g(b_+) \le 1\), we obtain

$$\begin{aligned} \begin{aligned} L&:= (a-b)^{p-1} (g(a_+) \eta _1^p - g(b_+) \eta _2^p) \\&= (a-b)^{p-1} ((g(a_+) - g(b_+)) \eta _1^p + g(b_+) (\eta _1^p - \eta _2^p)) \\&\ge (a_+-b_+)^{p-1} ((g(a_+) - g(b_+)) \eta _1^p + g(b_+) (\eta _1^p - \eta _2^p)) \\&\ge |G(a_+) - G(b_+)|^p \eta _1^p - p(a_+-b_+)^{p-1} |\eta _1-\eta _2| \max \lbrace \eta _1, \eta _2 \rbrace ^{p-1} \\&\ge |G(a_+) - G(b_+)|^p \eta _1^p - p |G(a_+)-G(b_+)|^{p-1} \\ {}&\quad \max \lbrace \eta _1, \eta _2 \rbrace ^{p-1} g'(a_+)^{-\frac{p-1}{p}} |\eta _1-\eta _2|. \end{aligned} \end{aligned}$$

By using Young’s inequality, Lemma A.3, and \(G(t)^p \le C (1+t)^{\gamma }\), we conclude that

$$\begin{aligned} \begin{aligned} L&\ge |G(a_+) - G(b_+)|^p \min \lbrace \eta _1, \eta _2 \rbrace ^p \\&\quad - \frac{1}{2^{2p-1}} |G(a_+)-G(b_+)|^{p} \max \lbrace \eta _1, \eta _2 \rbrace ^{p} - C \frac{1}{g'(a_+)^{p-1}} |\eta _1-\eta _2|^p \\&\ge \frac{1}{2^{p}} |G(a_+)\eta _1 - G(b_+)\eta _2|^p - C \max \lbrace (1+a_+)^{\gamma }, (1+b_+)^{\gamma } \rbrace |\eta _1 - \eta _2|^p, \end{aligned} \end{aligned}$$

where \(C = C(p) > 0\). \(\square \)

Let us finish the section with one more algebraic inequality.

Lemma A.6

Let \(a, b > 0\) and assume that \(\gamma < p\), \(\gamma \ne 0\). Then,

$$\begin{aligned} \max \lbrace a, b \rbrace ^{\gamma -p} |a-b|^p \le \left( \frac{p}{|\gamma |} \right) ^p |a^{\gamma /p} - b^{\gamma /p}|^p. \end{aligned}$$

Proof

We may assume without loss of generality that \(a > b\). Then,

$$\begin{aligned} |a^{\gamma /p}-b^{\gamma /p}|^p = \left| \int _b^a \frac{\gamma }{p} t^{\gamma /p-1} \,\textrm{d}t \right| ^p \ge \left| \frac{\gamma }{p} \right| ^p a^{\gamma -p} (a-b)^p, \end{aligned}$$

where we use the monotonicity of \(t \mapsto t^{\gamma /p-1}\). \(\square \)

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Kim, M., Lee, KA. & Lee, SC. The Wiener Criterion for Nonlocal Dirichlet Problems. Commun. Math. Phys. 400, 1961–2003 (2023). https://doi.org/10.1007/s00220-023-04632-w

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